1. Foundations of Constraint Processing All questions to Piazza
Lookahead Schemas
Foundations of Constraint Processing
CSCE421/821, Spring 2019
All questions to Piazza
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 360
Tel: +1(402)

2. Outline: Lookahead Schemas
Forward checking (FC)
Directional Arc Consistency (DAC)
Maintaining Arc Consistency (a.k.a. full arc-consistency)

3. Looking ahead: Rationale
As decisions are made (conditioning)
Revise the domain of future variables to propagate the effects of decisions (instantiations)
i.e., eliminate inconsistent choices in future sub-problem
Domain annihilation of a future variable avoids expansion of useless portions of the tree

4. Looking ahead: Techniques
Partial
forward-checking (FC)
directional arc-consistency (DAC)
Full
Maintaining arc-consistency (MAC) a.k.a. Real Full Lookahead (RFL)
Initialize the queue with the set of (Vf,Vc)
Revise(Vf,Vc), Vf future variable, Vc current variable

5. Revise(Vi,CVi,Vj)= Revise(Vi,Vj)
Revise the domain of Vi
Revising the domain of Vi given a constraint CVi,Vj on Vi (i.e., Vi  scope(C))
General notation: Revise(Vi,CVi,Vj)
In a binary CSP:
Revise(Vi,CVi,Vj)= Revise(Vi,Vj)

6. Revise(Vi, Vj) NOTE: only DVi may be updated
revised  nil
 x  Dvi
 y  DVj
If Check((Vi,x),(Vj,y)) Then break
revised  t
DVi  DVi \ {x}
Return revised
Simpler, equivalent code but not as obvious as the previous one

7. Domain filtering in lookahead
Vc current variable
Vf future variable
{Vf} all future variables
Revise(Vf, Vc)
FC(Vc):
 Vf  {Vf} connected to Vc
Revise(Vf,Vc)
If DVf ={} then break false

8. Directional Arc Consistency
Choose an ordering d, stick to it
After instantiating a variable at level i, do the following
For k from i to (n-1) in the ordering d Do
If FC(Vk)= false then break false

9. Real Full Lookahead (RFL)
AC-3
Set Q  {(Vf,Vc) | Vc is the current variable, Vf ∈ Neigh(Vf )}
Run AC-3(Q)
AC-1 does too many checks
First revise all the tuples in {(Vf,Vc)}
Then, call AC-1 ({(Vi,Vj), Vi,Vj are future variables}})

10. Look-ahead techniques: FC, DAC, RFL
assumes a fixed variable ordering d
RFL:
does more pruning (search may visit fewer nodes) at the cost of more consistency checks
FC(Vc)
FC(Vc);
While not failure: For the next Vf in the ordering d, FC(Vf)
FC(Vc); AC-3({Vf})
FC(Vc);
Repeat until quiescence or failure
 Vf1,Vf2  {Vf}, Revise(Vf1,Vf2)

11. Terminology overload alert: FC
FC is used to denote one of the following:
a partial look-ahead schema
a specific chronological backtrack search algorithm that uses the partial look-ahead schema
Meaning is inferred from context
Not a healthy situation, but a fact of reality
Advice: state upfront the meaning of your terms and stick to them throughout your paper

12. (BT Search +) RFL vs. FC
Reference: [Sabin & Freuder, ECAI94], [Bessière & Régin, CP97], [Sabin & Freuder, CP97], [Gent & Prosser, APES ], [Experiments by Lin XU, 2001], [Yang, MS thesis 2003]
Results: (sketchy)
Low tightness
High tightness
Low density (sparse) FC
RFL
High density (dense)
Note: Results depend on
Variable ordering (static vs. dynamic)
Problem difficulty (positive relative to crossover point)

13. Alert: Terminology Distinguish between MAC assumes binary branching
Real Full Lookahead (RFL)
Maintaining Arc Consistency (MAC)
MAC assumes binary branching